Just Hammond

This article features just intervals created by the mechanical tonegenerator of the classical Hammond B-3 Organ model.

Design of the Hammond B-3’s Tonegenerator

Since 1935 the Hammond Organ Company’s goal was to market electromechanical organs^[1] with 12-tone equally tempered (12edo) tuning. The mechanical tonegenerator of the Hammond B-3 Organ is based on a set of twelve different pairings of gearwheels that make (12*4) driven shafts turn. The corresponding driving gearwheels are mounted on a common shaft and turn all at the same rotational speed n₁. Certain gears reduce, others increase rotational speed.^[2]

For every chromatic pitch class four driven shafts are installed. Pure octaves are generated by dedicated tonewheels (with 2, 4, 8, 16, 32, 64 or 128 high and low points on their edges) that rotate with the driven shafts. Each high point on a tone wheel is called a tooth. When the gears are in motion, magnetic pickups react to the tonewheels’ passing teeth and generate an electric signal that can be amplified and transmitted to a loudspeaker.

For each pair of gearwheels the ratio of rotational speed n₂/n₁ is determined by the inverse ratio of the gearwheels’ integer teeth numbers Z₁ and Z₂:

[math]\frac{Z_1}{Z_2}=\frac{n_2}{n_1}[/math]

To calculate the rotational speed n₂ of the driven shafts we write

[math]n_2=\frac{Z_1}{Z_2}\cdot n_1[/math]

Table 1: Pairings of Gearwheels^[3] / Ratios and Intervals

Pitch Class	HAMMOND Organ Mechanical Gearing		Gear Ratio (canceled)			Conversion of raw ratio to interval	Deviation from 12tone equal temperament (12edo)	Intonation (Note A renders standard pitch, if shaft (A) is rotating @20 rev./sec)
	(A)	(B)	(C)	(D)		Conversion of raw ratio to interval	Deviation from 12tone equal temperament (12edo)
	driving Z₁[teeth]	driven Z₂[teeth]	Fraction		Ratio (C)/(D)	[cents]	[cents]	[cents]
C	85	104	85	104	0.8173077	-349.26	-49.26	-0.576
C#	71	82	71	82	0.8658537	-249.37	-49.37	-0.684
D	67	73	67	73	0.9178082	-148.48	-48.48	0.200
D#	105	108	35	36	0.9722222	-48.77	-48.77	-0.088
E	103	100	103	100	1.0300000	51.17	-48.83	-0.145
F	84	77	12	11	1.0909091	150.64	-49.36	-0.681
F#	74	64	37	32	1.1562500	251.34	-48.66	0.026
G	98	80	49	40	1.2250000	351.34	-48.66	0.020
G#	96	74	48	37	1.2972973	450.61	-49.39	-0.707
A	88	64	11	8	1.3750000	551.32	-48.68	0.000
A#	67	46	67	46	1.4565217	651.03	-48.97	-0.285
B	108	70	54	35	1.5428571	750.73	-49.27	-0.593

(Purple colored cells contain prime numbers)

Just Intervals

When we associate “ratios of the gearwheels’ integer teeth numbers” with “frequency ratios between partials” we realize an intrinsic just interval determined by integer teeth numbers within such mechanical gear - even without turning the shafts! Although the Hammond Organ pretends to generate a 12edo scale, the instrument in fact creates a high prime limit just scale.

Tuning

The whole set of frequency ratios is fixed by the design of the gear mechanism. The driving shaft’s (A) rotational speed n₁ determines the instrument’s (master-) tuning. Rotating at exactly 1200 rpm (20 rev./sec), the pitch of note A equals precisely 27.500 Hz or one of its doublings. Therefore the instrument aligns note A with a concert pitch of 440.0 Hz.

[math]f_\text{A}=20.0/\text{s}\cdot\frac{88}{64}\cdot(2^4)=440.0/\text{s} = 440.0 \text{ Hz}[/math]

Mapping Hammond’s Rational Intervals to the Harmonic Series

To find out, where the rational intervals played on a Hammond Organ occur in the harmonic series we

cancel the fractions of gear-ratios specified by Hammond and
calculate the least common multiple (LCM) of the denominators of "intervals of interest" by prime factorization
With this specific LCM we recalculate the numerators of the intervals. The resulting numerators correspond to the partial numbers we are looking for.

Before we proceed, we have to agree on a numbering scheme for octaves in the harmonic series.

Numbering Octaves

We apply the scheme from the article First Five Octaves of the Harmonic Series and number the octaves as follows:

Integer octave numbering starts with #1 for the range between the 1st and < 2nd partial

The 2nd octave starts at partial #2 (= 2¹) and covers partials 2 and 3
The 3rd octave starts at partial #4 (= 2²) and covers partials 4, 5, 6 and 7
The 4th octave starts at partial #8 (= 2³) and covers partials 8, 9, 10, 11, 12, 13, 14 and 15.
...

This numbering scheme is consistent with the scheme used by Bill Sethares^[4] : “In general, the n^th octave contains 2^n-1 pitches.”

Mapping Hammond’s Rational Intervals (cont.): Examples

The following examples illustrate how to map intervals or chords to the harmonic series.

Example 1: Mapping a single interval

In this example we map the combination of a Hammond Organ’s note E and a higher note A to the harmonic series.

Table 2: Mapping the fourth E-A

Pitch Class	HAMMOND Gear Ratio (canceled)		Prime Factorization					Ascending Partial Numbers of an Overtone Scale	Partial Found in Octave
	(C)	(D)	...of Column (D)					Recalculated Numerator = Partial# P=LCM *(C)/(D)	Counted from 1/1: Octave# = 1+(ln(P)/ln(2))
	Fraction (C)/(D)		...of Column (D)					Recalculated Numerator = Partial# P=LCM *(C)/(D)	Counted from 1/1: Octave# = 1+(ln(P)/ln(2))
E	103	100	2	2		5	5	206	8.7
A	11	8	2	2	2			275	9.6
Multiply --------> to find (D)'s least common multiple (LCM)			2	2	2	5	5	200 (LCM)	8.6 Decimal printed for orientation only

The resulting interval appears between partial # 206 and partial # 275. Thus the frequency ratio is (275:206), which equals 500.14 cents.

Example 2: Mapping a chord

Adding an upper fifth (note B), the second example illustrates how to map the resulting sus4-chord E-A-B to the harmonic series.

Table 3: sus4-chord E-A-B

Pitch Class	HAMMOND Gear Ratio (canceled)		Prime Factorization						Ascending Partial Numbers of an Overtone Scale	Partial found in Octave
	(C)	(D)	...of Column (D)						Recalculated Numerator = Partial# P=LCM *(C)/(D)	Counted from 1/1: Octave# = 1+(ln(P)/ln(2))
	Fraction (C)/(D)		...of Column (D)						Recalculated Numerator = Partial# P=LCM *(C)/(D)	Counted from 1/1: Octave# = 1+(ln(P)/ln(2))
E	103	100	2	2		5	5		1442	11.5
A	11	8	2	2	2				1925	11.9
B	54	35				5		7	2160	12.1
Multiply --------> to find (D)'s least common multiple (LCM)			2	2	2	5	5	7	1400 (LCM)	11.4 Decimal printed for orientation only

The supplemental note B establishes an additional prime factor. We find the matching pattern of partials for this sus4-chord (1442:1925:2160) farther up in the harmonic series, where this chord spans the boundary between the 11^th and the 12^th octave.

Example 3: Mapping all of the tonegenerator's pitchclasses

The full set of the Hammond Organ’s intervals resides surprisingly far up in the Harmonic Series:

The 44th octave starts at partial #(2⁴³), just below the set of partials determined by the Hammond Organ’s tonegenerator
The 45th octave starts right within the derived set of partials and starts at partial #(2⁴⁴)

Table 4: The full set of intervals' position in the Harmonic Series

Pitch Class	HAMMOND Gear Ratio (canceled)		Prime Factorization																Ascending Partial Numbers of an Overtone Scale	Partial Found in Octave #
	(C)	(D)	... of Column (D)																Recalculated Numerator = Partial # P=LCM *(C)/(D)	Counted from 1/1: Octave# = 1+(ln(P)/ln(2))
	Fraction (C)/(D)		... of Column (D)																Recalculated Numerator = Partial # P=LCM *(C)/(D)	Counted from 1/1: Octave# = 1+(ln(P)/ln(2))
C	85	104	2	2	2									13					15,003,356,791,500	44.8
C#	71	82	2														41		15,894,517,438,800	44.9
D	67	73																73	16,848,249,818,400	44.9
D#	35	36	2	2				3	3										17,847,130,301,000	45.0
E	103	100	2	2						5	5								18,907,759,758,888	45.1
F	12	11											11						20,025,870,883,200	45.2
F#	37	32	2	2	2	2	2												21,225,337,107,975	45.3
G	49	40	2	2	2					5									22,487,384,179,260	45.4
G#	48	37														37			23,814,549,158,400	45.4
A	11	8	2	2	2														25,240,941,425,700	45.5
A#	67	46	2												23				26,737,439,929,200	45.6
B	54	35								5		7							28,322,303,106,240	45.7
Multiply --------> to find (D)'s least common multiple (LCM)			2	2	2	2	2	3	3	5	5	7	11	13	23	37	41	73	18,357,048,309,600 (LCM)	45.1 Decimal printed for orientation only

Discussion

No doubt - the evidence that a cluster of 12 simultaneously ringing semitones from a Hammond Organ is allocated around the 45^th octave of the harmonic series is of limited practical value. The intervals' far-up placement is mainly caused by Laurens Hammond’s implementation of various prime numbers (11, 13, 23, 37, 41, 73) in different gearwheel pairings.

Respective high-order partials are very densely spaced (in the range of pico-cents) and intervals between successive partials up there are too narrow for musical applications by far

Due to its construction the tonegenerator selects only twelve from 17.6 trillion varieties in the 45^th octave where…
- the partial number associated with the LCM, which is located exactly 8/11 below pitch class A, is not addressed because there is no gear with transmission ratio 1.000
- no pure octave above a virtual root (1/1; partial# (2⁴⁴)) is playable, which would ring -624.997 cents way down from pitchclass A

General Applicability

The method of prime factorization to find the LCM can be applied to arbitrary intervals, chords or scales built from rational intervals to identify their position in the harmonic series. Simply replace the gear-ratios by just intervals of interest.

References

↑ Webressource https://en.wikipedia.org/wiki/Hammond_organ (retrieved December 2019)
↑ Detailed photos of a similar M-1 tonegenerator are provided by https://modularsynthesis.com/hammond/m3/m3.htm (retrieved December 2019)
↑ Gearing details were taken from http://www.goodeveca.net/RotorOrgan/ToneWheelSpec.html (retrieved Dec 29, 2019) The German Wikipedia provides the same technical information (in German): https://de.wikipedia.org/wiki/Hammondorgel#Tonerzeugung (retrieved Dec 29, 2019) The HammondWiki publishes a second, alternative set of gear ratios with slightly deviating pitch class “E”. Certain other pitch classes are shifted by pure octaves. http://www.dairiki.org/HammondWiki/GearRatio (retrieved Dec 29, 2019)
↑ Sethares, William A. Tuning Timbre Spectrum Scale. London: Springer Verlag , 1999.